Related papers: Diophantine Tuples over $\mathbb{Z}_p$

In this paper, we define a $k$-Diophantine $m$-tuple to be a set of $m$ positive integers such that the product of any $k$ distinct positive integers is one less than a perfect square. We study these sets in finite fields $\mathbb{F}_p$ for…

For a nonzero integer n, a set of m distinct nonzero integers {a_1,a_2,...,a_m} such that a_i a_j + n is a perfect square for all 1 <= i < j <= m, is called a D(n)-m-tuple. In this paper, by using properties of so-called regular Diophantine…

A Diophantine $m$-tuple is a set of $m$ distinct integers such that the product of any two distinct elements plus one is a perfect square. It was recently proven that there is no Diophantine quintuple in positive integers. We study the same…

For a prime p, a Diophantine m-tuple in $\mathbb{F}_p$ is a set of m nonzero elements of $\mathbb{F}_p$ with the property that the product of any two of its distinct elements is one less than a square. In this paper, we present formulas for…

For a nonzero integer $n$, a set of $m$ distinct nonzero integers $\{a_1,a_2,...,a_m\}$ such that $a_ia_j+n$ is a perfect square for all $1 \leq i < j \leq m$, is called a $D(n)$-$m$-tuple. In this paper, we show that there infinitely many…

For a nonzero integer $n$, a set of distinct nonzero integers $\{a_1,a_2,\ldots,a_m\}$ such that $a_ia_j+n$ is a perfect square for all $1\leq i<j\leq m$, is called a Diophantine $m$-tuple with the property $D(n)$ or simply $D(n)$-set.…

Let $K$ be an imaginary quadratic field and $ \mathcal{O}_K$ be its ring of integers. A set $\{a_1, a_2, \cdots,a_m\} \subset \mathcal{O}_K\setminus\{0\}$ is called a Diophantine $m$-tuple in $\mathcal{O}_K$ with $D(-1)$ if $a_ia_j -1 =…

A set of $m$ distinct nonzero rationals $\{a_1, a_2,\ldots, a_m\}$ such that $a_i a_j+1$ is a perfect square for all $1\le i <j \le m$, is called a rational Diophantine $m$-tuple. If in addition, $a_i^2+1$ is a perfect square for $1\le i\le…

A set of $m$ positive integers $\{a_1, a_2, \dots , a_m\}$ is called a Diophantine $m$-tuple if $a_i a_j + 1$ is a perfect square for all $1 \le i < j \le m$. In 2004 Dujella proved that there is no Diophantine sextuple and that there are…

Let $n$ be a non-zero integer. A set $S$ of positive integers is a Diophantine tuple with the property $D(n)$ if $ab+n$ is a perfect square for each $a,b \in S$ with $a \neq b$. It is of special interest to estimate the quantity $M_n$, the…

A rational Diophantine $m$-tuple is a set $\{a_1,\ldots,a_m\}$ of distinct nonzero rational numbers such that $a_i a_j+1$ is a square for all $1\leq i < j\leq m$. Similarly, we may ask when $a_ia_j+1$ is a $k$-th power. Here, we study the…

A set of $m$ distinct nonzero rationals $\{a_1,a_2,\ldots,a_m\}$ such that $a_ia_j+1$ is a perfect square for all $1\leq i<j\leq m$, is called a rational Diophantine $m$-tuple. It is proved recently that there are infinitely many rational…

A Diophantine $m$-tuple is a set $A$ of $m$ positive integers such that $ab+1$ is a perfect square for every pair $a,b$ of distinct elements of $A$. We derive an asymptotic formula for the number of Diophantine quadruples whose elements are…

Let $(a_1,\dots, a_m)$ be an $m$-tuple of positive, pairwise distinct, integers. If for all $1\leq i< j \leq m$ the prime divisors of $a_ia_j+1$ come from the same fixed set $S$, then we call the $m$-tuple $S$-Diophantine. In this note we…

A Diophantine $m$-tuple over a finite field $\mathbb{F}_q$ is a set $\{a_1,\ldots, a_m\}$ of $m$ distinct elements in $\mathbb{F}_{q}^{*}$ such that $a_{i}a_{j}+1$ is a square in $\mathbb{F}_q$ whenever $i\neq j$. In this paper, we study…

Let $k\geq 2$ and $n\neq 0$. A Diophantine tuple with property $D_k(n)$ is a set of positive integers $A$ such that $ab+n$ is a $k$-th power for all $a,b\in A$ with $a\neq b$. Such generalizations of classical Diophantine tuples have been…

Given a finite set of primes $S$ and a $m$-tuple $(a_1,\dots,a_m)$ of positive, distinct integers we call the $m$-tuple $S$-Diophantine, if for each $1\leq i < j\leq m$ the quantity $a_ia_j+1$ has prime divisors coming only from the set…

For an integer n, a set of m distinct nonzero integers {a_1,a_2,...,a_m} such that a_i a_j+n is a perfect square for all 0<i<j<m+1, is called a D(n)-m-tuple. In this paper, we show that there are infinitely many essentially different…

We consider Diophantine quintuples $\{a, b, c, d, e\}$. These are sets of distinct positive integers, the product of any two elements of which is one less than a perfect square. It is conjectured that there are no Diophantine quintuples; we…

A set of m distinct positive integers {a_{1},...a_{m}} is called a Diophantine m-tuple if a_{i}a_{j}+n is a square for each 1\leqi<j\leqm . The aim of this study is to show that some P_{k} sets can not be extendible to a Diophantine…